Our Democracy is hampered by a voting system that poorly reflects the will of its electorate, one that often forces us to choose between the lesser of two evils. But there are alternative systems, approval voting and score voting, which can do the job better.

Tuesday, May 3, 2011

The Alternative Vote: Neither the Death nor Resurrection of Democracy

This Thursday, the UK is holding a referendum on voting. Specifically, voters are being asked whether they want to stick with the status quo--"first past the post" (FPTP) or "plurality" voting--or to try something new. The new method they are considering goes by the name "the alternative vote" (AV), but it's known in the US as "instant runoff voting" (IRV) or "ranked choice voting" (RCV), and unless you're new here you've heard me heap criticism upon it. For this article, I'll be using the UK terms.

Current polls show the measure is teetering towards failure, which if fine from my perspective, but both the "Yes" and "No" campaigns have been flying some pretty ridiculous propaganda, so let's set the record straight. AV will not cause democracy to crash and burn. They've used it for a hundred years in Australia, and it hasn't turned them into an arid wasteland of venomous monsters (well, no more so than it already was.) Nor will it create a utopia of perfect democracy. Again, Australia has used it for a hundred years, and everyone complains about the government just as much as they do in any other free country in the world.

In practice, AV gives about the same results as FPTP. I've seen a claim that over 100 seats would have been allocated differently, but that's based on counting any seat where the winner had less than 50% of the votes as a potential switch; the actual number, based on reasonable assumptions about voter's second- and later-choices, is 3, or 1%.

Furthermore, with perfectly-tactical voters, AV always gives exactly the same results as FPTP: It is only beneficial to the degree that voters will choose honesty over tactics, and despite the "Yes" campaign's claims, AV voters still can benefit from, and so will be incentivized to practice, tactical, rather than honest, voting. This is trivial to show, especially in the situation the UK finds itself in now, with three strong parties that have no natural and obvious split among them as a basis to form a coalition. While it is true that AV makes it more-difficult for a small party to act as a spoiler--which is the kernel of truth that the "Yes" campaign has built this fabrication around--it has the same susceptibility to spoilers as FPTP when there are three or more strong parties, and it's fear of spoilers that drives tactical voting. (AV can still result in hung parliaments too, for which I again point to Australia, who saw its two largest blocs walk out with 48% of seats each in their 2010 election.)

While AV does, in a limited way, shore-up this one weakness in FPTP, it also introduce its own special failure modes, including non-monotonicity, participation failure, and other paradoxes of voting which cannot occur under FPTP, but which are quite common under AV. The improvements are small, and then almost entirely countered by these new failures. Failures which commonly are used as the basis for repeal campaigns by voters who feel they've been sold a false bill of election-reforming goods. (Expect the repeal movement to peak after two or three elections have passed under AV.)

But even if we follow the naive notion that every voter would choose honesty, and that the marginal improvements are worth the new paradoxes (and that the inevitable repeal movement can be fought back), then AV still comes at a steep cost. Ballot-spoilage rates are four to seven times higher with AV. Not, as the "No" campaign has said, because voters can't understand the process, but simply because there are so many ways to make a ballot-invalidating error when filling out such a ballot. Perhaps related, the counting process is notably more expensive; not astronomically so, as the "No" campaign has claimed, but it will require either the purchase and maintenance of more complex, and therefore more costly, voting machines, or a more time-consuming, and therefore more costly, amount of hand-counting.

The UK does have a few points in its favor that would make AV less of a wasted effort than it would be, for instance, in the US. The relative lack of effective political polling in the UK leaves voters with less information with which to make informed tactical voting decisions, so "honesty" may be a somewhat more-likely default than it would in the US. And since UK voters do not directly choose the head of government, a unitary office, there is less of central figure for voters and parties to rally, and become divisive, around. Both of these aspects increase the likely amount of improvement in outcomes due to AV, from "minuscule" to "tiny". However even these mitigating trends are on the wane, with political polling catching on and party heads taking a more active role as the faces of their parties.

Finally, there is the notion that a vote on this (non-)reform sends a message. The "Yes" campaign says that a yes vote sends the message that you want something to be done and this is the first step towards proportional representation, while the "No" campaign says that a no vote sends the message that this is not the reform you want and that you would rather have proportional representation. Uh huh. My advice is that your vote for or against AV should be based solely on the (lack of) merits of AV, irrespective of what better "sends the message" that you really want proportional representation (PR). The fundamental prerequisite for PR is multimember districts, and no government anywhere has ever enacted AV and then later been convinced to add multimember districts. The two reforms are essentially unconnected.

AV isn't worth the effort. I would encourage UK voters to vote no on AV, and to campaign for PR.

36 comments:

Well put piece. Still, for newcomers (like myself), you may want to better explain terms like on-monotonicity, and participation failure. Most UK voters who may stumble upon this entry probably won't know what those terms mean either.

As for whether IRV will create the results the proponents are claiming, you're probably right it won't,but I'm not so sure Britain should vote "no" on it, because it may very well send the message the Brits don't want electoral reform. It's hard to say, because while it may not lead to an evolution of PR, a no vote may create a discouragement between PR activists/advocates that most Brits want the "system as it is". But since I'm not British it's not up for me to say.

I think the Lib Dems were at first for some form of PR upon entering in a coalition with the Tories (STV maybe? I don't know), but "compromised" to a mix of AV and PR, then just AV. I think they showed a serious weakness playing all their cards on a weak reform like IRV...which probably won't grant them as many new "Lib Dems" as they think, with an angry party base at all their recent betrayals.

Anyway, I'm blabbing on, and I apologize. Do you have a particular form of PR you'd recommend to Britain, and/or Range/Approval voting?

Scratch out the "but I'm not so sure Britain should vote "no" on it, because it may very well send the message the Brits don't want electoral reform.", I was tired and high on oxycodone (with a legally acquired prescription of course..) so I wasn't thinking so straight. In the end you're probably right, IRV isn't so hot, and while I view the events in Britain with a level of detachment (since well, I'm not British), I do hope they strive for better electoral reform in the future. It'll be interesting to see the results that if it wins though, and how it effects electoral reform in the United States (And Canada?)

I'd hoped to go back and include links to describe things like non-monotonicity, but time flies, and with the vote on Thursday, I let it go out un-linked. Perhaps I'll get some time to edit it today.

The LibDems did, indeed, want STV, and AV is just a "miserable little compromise".

Any form of PR is a huge step. STV has worked fairly well--again, Australia, where it's used to elect the Senate--but Germany's system (half by single-member plurality, the other half filled by party-list to achieve proportionality) also seems quite effective.

Personally, since I'm such an approval/range proponent and have a dislike of party-list methods, I rather like the idea of re-weighted range (or re-weighted approval) as a PR method.

DLW, honestly, I'd rather have just one system; at most one system per chamber. 3 of one and 1 of another seems just... off.

I think the Lib Dems made a huge mistake from the beginning picking the Cons as their coalition partners. Then again everyone hated Brown, so I guess it was a sour pick either way. And now, with all their betrayals, AV I don't think is going to give them the return on investment they are looking for. We'll see though I suppose.

STV it seems has worked out well where it's used (The Senate in Australia seems better than the House, at least that's what I gather from the Aussie's I talk to). Germany's MMP system is pretty good as well, if satisfaction is any measure.

Myself, I love the idea of PR, but *hate* (ok maybe not hate, but dislike) partylists, I'm of course willing to compromise that, and admit partylists can work out, but I wish it didn't *have* to be that way. Admittedly, I know nothing about re-weighted range or approval, I should probably read up more on the Range/Approval voting methods period. Advocates in the past though have turned me off to it...

Yes, approval/range advocates are more often mathematicians or programmers than diplomats... friendly persuasion isn't what we're known for (I'm a programmer myself, but I like to think I'm one of the more friendly and persuasive ones).

If you haven't already checked it out, http://rangevoting.org/ is a wealth of knowledge (but its very much written by a mathematician.)

I've always been drawn to this graphic in particular: http://rangevoting.org/BayRegsFig.html

And here is an explanation of re-weighted range (which you can convert to re-weighted approval by restricting the range to {0,1}): http://rangevoting.org/RRV.html

You're definitely one of the more friendly ones, that is for sure. Perhaps though, the Range/Approval voting movement(?) should adopt some more diplomatic people to lead, or at least be personable talking heads. In the realm of politics, diplomacy is always a necessity, at least while you are building up your army of support.

My point is that folks don't like party-lists and STV is complicated and has the same issues as IRV due to the easy proliferation of permutations and what-not.

So low-numbered LR-Hare keeps the lists tacit and lowers the barrier to third parties being able to win a seat and thereby gives more voice to ethnic/economic/ideological minority groups.

But since LR Hare tends to smaller parties more so, it needs to be checked by using a single-seated election that tends to favor bigger parties more so. Now, I'm flexible over which election rule is used for said election, but I'll admit that Approval Voting likely would be very (if not most, apart from range-voting) effective at making this election be competitive.

I personally think there might be great synergies from combining different types of elections, like with two(or three) stage elections.

Needless to say, my approach would put a premium on the top party winning two of the 3 LR Hare seats in a a majority of the districts (and at least one of the 3-seats in the other districts) and then a majority of the single-seated elections to get control of a majority of the seats.

I'm guessing that FPTP would make it easier for one party to win a majority of the single-seated elections so Approval Voting would probably increase the likelihood of coalition government. However, if there were two bigger parties and many smaller parties then the party with a clear plurality could mitigate the bargaining power of their smaller potential coalition members by playing them off of each other.

IDK, you see I don't care about nailing proportionality in representation. For me, what matters is to make the elections competitive and the bigger parties more responsive to the issues of more people, while maintaining the hierarchy that I think is inevitably needed for political leadership.

>But I do happen to think that nailing proportionality would go a looong way toward making parties responsive to the people.<

Perhaps a mix like Germany has to a degree? Part PR (STV, or re-weighted approval/range or X other PR system), part single seated Approval/Range? It's probably best to experiment with differing methods before we commit to one rule for all elections.

proportionality in representation does not equal proportionality in power at any particular time and place. Thus, if the system were to tend towards proportionality in representation and power over time, with some of the tendency due to shifting priorities of existing parties, then that would suffice.

According to "choosing an electoral system", the biggest predictor of proportionality in representation immediately after an election is the number of seats contested. But increasing the number of seats contested tends to decrease the competitiveness of the elections, due to the predictability of the results. This is why I don't think nailing proportionality is a worthy goal.

That and I also believe that elevating people's habits as citizens trumps electoral reform. Yet I also believe the use of more multi-seated "more local" elections would encourage the proliferation of LTPs that'd be more effective at making more people better citizens due to the diseconomies of scale in building community.

Economics often uses ordinals because it's harder to measure, in the purely abstract sense, "how much better?" than it is to measure "better or worse?" and this is true in all aspects of theoretical mathematics, all the way down to set theory.

Luckily, I'm an engineer, not a theoretical mathematician ;)

You can convert everything to money ("But would you rather have an apple, or an orange AND $3?") but even then, the marginal utility of money isn't constant (having another $20 is a lot more important to me than it is to Warren Buffet.) Money is a _better_ analog to true utility than any physical good is, but it's still not perfect. And there is nothing perfect.

And so, as an abstract concept, cardinal utilities are hard to measure in the real world and pretty much impossible to compare between individuals.

And that's okay.

Because the use of cardinal values wasn't the GOAL for range voting's advocates; it just turned out that, after constructing a model in which to test voting systems (one in which model voters were *assigned* true utilities, side-stepping the measuring hurdle) range voting happened to perform surprisingly well. You have to remember that the simulation was done FIRST, and only AFTER seeing its results did anyone become an advocate for range voting. Why would they, there had been NO theoretical work done on ordinal methods, because it was too hard to do anything with it using constructive proofs. Many mathematicians HATE probabilistic methods (you can't even call it probabilistic "proof") and that's essentially what the voting simulation was. It's not beautiful... but it works. (See? Engineer, not mathematician.)

Even a highly imperfect tracker of true utilities, like range voting, turns out to provide a much better algorithm for elections than more-cleanly-provable ordinal system.

So then you'd agree that if someone thought assigning model voters *true* utilities wasn't adequate to judge election rules that they'd be consistent in concluding that there still remains an electoral reform debate?

I wonder what if in addition to modeling voter preference one randomly simulated a parameter for a family of monotonic transformations of voter preferences and then determined the votes on that basis. So let's say that f[x, a]= 10^(1-a)*x^a where 0<=x<=10 and 0<a with a determined randomly by a pdf with a mean of 1. Then you could assess the outcomes based on the Bayesian Regret with a=1.

I wonder what that would do? The way I see it, the transformations wouldn't affect the FPTP or ranked voting rules but it would affect Approval-Score voting, albeit it's impact would be more on the variability of the voting outcomes than their mean BR value.

As you know, I'm not against Approval Voting so much as I'm skeptical about arguing for its superiority on the basis of Bayesian Regret analysis. It's a great thought experiment but is it the smoking gun? I don't think so...

Well, sure; if you reject the assumptions of a model, you could easily reject its conclusions.

But let's take a look at what you're rejecting: that voters have true, comparable, utilities, and that those utilities are distributed in some way that we can reasonably approximate and simulate with a computer. Is that really something that is controversial?

Now you may ask "in what way are the distributed?", and there are several possible answers to that, which is why several different distributions (and different parameters for those distributions) were tried. And it's not as if range and approval were a bit better under a few different distribution/parameter settings: they were ENORMOUSLY better under ALL of them. And different models of strategy (which seems like it could be an analog to your monotonic transformations if the different distributions/parameters don't cover it?) were also tried, and range and approval were still enormously better under all conditions.

The only thing that hasn't been seriously examined (the CRV mailing list was talking about this last week) is if there is a candidate-biased correlation with honesty, could that be detrimental? (In other words, if voters who like candidate A are much more likely to vote strategically than voters who like candidate B, what does that do to BR for all the various voting systems?) I suspect that range and approval will continue to out-score all other methods, but we can't really say for sure until we try it out.

Another thing: for some of the different distribution models, it is possible to solve them (in the traditional constructive sense) and make predictions like: in a three candidate race, how often does IRV suffer from non-monotonicity? And then we can compare those predictions with the rate of real-world occurrences (lots of math). Of the three models examined, one was a little too high, one was a little too low, and the third was pretty close to the observed value. And remember, under ALL THREE of those models, range and approval had the best performance. And so I think it is very unlikely that anyone will be able to create a model that is as accurate to the real-world as any of these three while ALSO concluding that something other than range or approval is the best-performing method. It could happen, but I wouldn't bet on it.

Really, I think we have ENOUGH evidence (even if it's not 100% iron-clad conclusive evidence) to say "It's time for some real-world trials of approval and range." We've done, I think, as much as we can with simulation; and the simulation has made an INCREDIBLY strong statement about what to do next.

"Really, I think we have ENOUGH evidence (even if it's not 100% iron-clad conclusive evidence) to say "It's time for some real-world trials of approval and range." We've done, I think, as much as we can with simulation; and the simulation has made an INCREDIBLY strong statement about what to do next. "

Is there any organized attempt to do so? I'd figure some local efforts may be best (maybe focusing on localities that utilize local initiatives &/or referendums?)

One thing I've noticed as an outsider in regards to electoral reform is the Range/Approval voting advocates have horrible decorum, and generally come off as obnoxious trolls. Presentation is important and I think the Range/Approval voting advocates really need to work on that (If they already are, my apologies).

Say what you want about Rob Richie, but from what I've gathered, he's a suave motherfucker who been able to lead a movement that's finally getting larger traction (Maine in particular is increasingly considering IRV for state wide races). If I were to give any advice (not that you'd give a shit about some nobodies advice over the interwebs...), maybe you should meet him at his game, rather than solely relying on mathematics to do the talking.

If I understand the links, it seems like any constructive proof can be disproved by the provision of a counter-example.

By randomly generating for each voter a set of valuations for the candidates and an "a" that determines the transformation given to their preferences, it might be possible to come up with a case where Approval Voting doesn't come in first. In this case, it may be necessary to modify the Bayesian Regret function some. In Economics, we model how we deal with risk often by means of a Mean-Variance analysis. I think there might be a Mean-Variance analog with Bayesian Regret so that the variability of the election outcomes would also matter for the desirability of an election rule.

My goal is to put some balance into the use of the Bayesian Regret tool. My dictum is that if you lower your expectations then you might be surprised. What if BR is very useful to draw out the possible/likely implications of different election rules, rather than purportedly identifying the "right" election rule.

Let me explain why a Mean-Variance objective function might be a good idea. Most major changes in gov't take a while to work out and so if you have very strong changes in who is in power then it would be hard to make serious changes. For each new party(or party-coalition) in power could easily throw a wrench in development of the projects set in motion by its predecessor. dlw

Dale:"voters have true, comparable, utilities, and that those utilities are distributed in some way that we can reasonably approximate and simulate with a computer. Is that really something that is controversial?"

dlw: 1. true utilities are seen by some as like round squares, a postulate that is so invalid that even God cannot prove it.

3. It is not correct to describe what you are doing as approximating. It's just simulations.

"different models of strategy (which seems like it could be an analog to your monotonic transformations if the different distributions/parameters don't cover it?) were also tried, and range and approval were still enormously better under all conditions."

dlw: I'm not presuming different strategies. I'm presuming different interpretations of the ordinal utilities assigned to voters. It only takes one counter example to disprove a constructive proof and my intuition is that relaxing the cardinality assumption via the transformations and valuing both the closeness to the ideal and variability of the outcomes would subvert the purported superiority of AV/SV on the basis of BR analysis.

I also agree with TF that marketing and manners matter. The hubris that can come from mathematical modelling can lead to failure. I have also met Rob Richie in person here in MN and he was very polite to me even though I was pretty much saying the same thing: why are you pushing IRV so hard when what we really need are more multi-seated "more local" elections?

Other constructive projects might be to compare how far apart AV and IRV are when there are only 3 candidates. Or how would AV3(only can approve of up to 3 candidates) do compared with AV. Or to see how much IRV3 can be improved upon by using AV3 for the first stage. As an engineer, I'd think these sorts of questions might be appealing for you. You don't get to pick all aspects of the options available. You need to make the best of what seems plausible and an IRV3/AV3 seems far more plausible at this point due to the success of FairVote at marketing and manners with US voters.

TF,It's hard to be " purely constructive", since often there must be a process of creative destruction.

But I will admit I'm entertaining the idea of local rules, meaning what if a system like the UK let each district choose its own rule for electing one-four MPs each election year. Then there'd be a process of learning by doing about which of the wide variety of election rules works best. And advocates for election rules that are promising but not well known would have a better chance to get their rule adopted somewhere.

TF, Speak for yourself. The only reason I'm an electoral reform junky is to feed my ego.;-)

Dale, I posted over at my blog about "local election rules". This would be a way to make Approval Voting get used without appealing to BR simulations. We could have a real laboratory experiment and lots of learning by voting...

TF: There have been some groups working toward approval voting, like FairVote has been working towards IRV. There was a strong push in Colorado by a Libertarian-leaning group of people that, unfortunately, seems to have mostly fizzled after their gubernatorial candidate's terrible performance, and there was actually a bill proposed in New Hampshire (by a Republican) but it was shot down (by another Republican that had shot down a similar bill 30 years earlier and is still on the elections committee.)

Also, a group based out of the Center for Range Voting has been working towards a 501(c) group in California to work towards approval and range voting; still lots of paperwork, but it's happening! They're calling it the Center for Election Science.

DLW: I do have concrete answers for some of your questions; for instance, in three candidate elections, approval and irv differ on about 4% of outcomes (using the random-normal election model); approval selects the "best" winner about 74% of the time, and irv about 70% (for comparison, plurality gets about 66% and range about 78%, so there's a pretty regular 4% improvement for each step going plurality->irv->approval->range).

Your point about variance is a good one, but since approval and range, in 3 candidate races, when they miss the best choice usually get the 2nd, while plurality and irv when they miss (which is more often) usually get the 3rd, the variance will also be better with approval and range (irv and plurality both miss more often and when they miss they miss by more).

Philosophical criticisms of utilitarianism... what can I say that hasn't been said for 100 years? But I do think that much criticism of it has hinged around the inherent unknowability of true utilities, which the simulations avoid by random assignment.

Finally: a simulation IS an approximation. If you follow some of the math links I provided, you'll see where Dr. Smith runs a simulation, and also computes an exact value for a result; the simulation is an approximation (and he usually gets to within 0.1%). Then he'll use the simulation to get numbers that he CAN'T get precise numerical values for. But, meh, that's semantics. ;)

Ideas: Compliment IRV as best for elections with fewer candidates and less reliable when the number of candidates proliferate or when there are three strong candidates!

It would be interesting to compare an IRV3/AV3 with an AV3 or AV with say the typical number of candidates for an election. If IRV works best with 3 candidates then it'd work even better if it were used with the three candidates who were approved the most.

The variance thing needs to be coupled with the impact of relaxing the cardinal utility assumption so that there's more indeterminancy about how many candidates are approved by the voter. If ln(a) has a standard normal distribution then it's typical range would be between -2 and 2 and a would be between e^-2 and e^2 or .14 and 7.39.

So let's say that someone's utility for 4 candidates is 2.5, 5, 6, 7.5. Then how would they vote with an approval voting rule? Then how would they vote with an a of 3 so their "utilities" would be .15625, 1.25, 2.16 and 4.21875?Likewise what if a=.2 so we'd get 7.58, 8.71, 9.03, 9.44?

This is my point. All of your simulations have been done with the presumption of cardinal utility. Once this is relaxed then it seems plausible that there would be an inherent indeterminancy in how someone would vote with an Approval or Range voting rule, which would make its outcomes more variable.

Dale, I'm interested in a simulation that could capture the predicted percent of times a 3-seated election with a Hare Quota will result in a non-majority "rule" (or a coalition between 2 of the 3 who did not get a majority of the vote.). I'm thinking that if the majority of the time, a majority still wins that averaged over many mega-districts it can be more or less assured that a majority will rule...

Like for any voting system, there's no algorithm that doesn't rely, in some way, on information about how other voters will behave. And without perfect information, none of the algorithms will be perfect.

A good starting place would involve identifying who the two most-likely-to-win candidates are; identify your favorite of those, approve of them and all you like better.

But strategies would differ on what to do for any candidates that fall between your favored and dis-favored of the top-two, or what your best strategy would be if it's a close contest for 2nd-most-likely-to-win (i.e., a true three- or more-way contest.) For that, you'd need a more involved model, that gives good estimates for each candidates chance to win, and you'd base your approval on whether or not approving that candidate increases your expected utility. (Basically, "how low (on my utility) can I go before I cause to be elected someone worse than I could have?")

There's a droop quota and a hare quota. The droop quota guarantees a majority rule and favors somewhat bigger parties. The hare quota makes the outcome more proportional and favors somewhat lesser parties.

I'm arguing that if you only have 3 contested seats that it already favors the bigger parties enough that a Hare quota should be considered and want to simulate how likely it would be for a non-majority to be in power if say 3-seated STV with a Hare Quota were used for like 67 state legislative elections.dlw

(Hmm... actually, I wonder if you would get better average performance by picking 2 candidates at random and approving all candidates above the median utility of just those 2.)

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Oh, right; Hare. But wouldn't that just make it worse? Each winner got 1/3rd of the vote, so any two members will have 2/3rds of the vote, which is clearly more than 50%?

The problem is "majority" isn't something that's clearly defined for a multi-winner system (particularly for a non-party-list based one.) So I still don't know what you mean by "a coalition who did not get a majority of the vote." Precisely define this, or at least give me a pair of examples, one where a majority wins and one where it loses, because I honestly have absolutely no idea what you think that means.

DHS:Oh, right; Hare. But wouldn't that just make it worse? Each winner got 1/3rd of the vote, so any two members will have 2/3rds of the vote, which is clearly more than 50%?

dlw: No, because a candidate locks in their seat with 1/3rd of the vote and the surplus votes above 1/3rd rather than 1/4th would be tranfeerred. After the candidates with more than 1/3rd of the votes are removed then the remaining seat(s) are allocated based on who's got the most top-ranking votes after systematically eliminating all but two of the candidates.

DHS:The problem is "majority" isn't something that's clearly defined for a multi-winner system (particularly for a non-party-list based one.) So I still don't know what you mean by "a coalition who did not get a majority of the vote." Precisely define this, or at least give me a pair of examples, one where a majority wins and one where it loses, because I honestly have absolutely no idea what you think that means.

dlw: Okay. You are right. I guess one could just take the pseudo party-affiliation to be the top-ranked candidate for a voter among the five finalists in a 3-seated election. The idea is to simulate this for 67 three-seated elections and then use some simple rules for determining the ruling party coalition and see how often the ruling parties get less than 50% of the vote. I'm guessing that there'd also need to be some simulation of the entry of candidates into the election, which of course would add additional ad hoc assumptions...

All of that to try and test an intuition that the possibility of a non-majority rule with a Hare quota does not mean that such happens a majority of the time or would be likely with very many 3-seated elections...dlw

I'm still mulling over the rest of your thoughts; but yes, the algorithm used for Dr. Smith's BR simulations was "approve all candidates above the median utility of all candidates."

So if the utilities were .1, .2, .3, and .9, the median is 0.375, and only the .9 candidate is approved.

This algorithm has the advantage of always approving at least one candidate, and always not approving at least one candidate, but being able to approve the full range in between, depending on how the utilities are distributed. It's also pretty easy to calculate.

I swear Smith did some examination of different approval strategies, but I can't find the page for it now :/ I seem to recall that this method wasn't the best, but it was pretty good.

DSH: the algorithm used for Dr. Smith's BR simulations was "approve all candidates above the median utility of all candidates."

So if the utilities were .1, .2, .3, and .9, the median is 0.375, and only the .9 candidate is approved.

dlw: I believe the median would be .25. The mean is .375. Which is it the mean or the median?

DSH: This algorithm has the advantage of always approving at least one candidate, and always not approving at least one candidate, but being able to approve the full range in between, depending on how the utilities are distributed. It's also pretty easy to calculate.

dlw:If it's the mean, the sorts of transformations I described could make a difference.

dsh:I swear Smith did some examination of different approval strategies, but I can't find the page for it now :/ I seem to recall that this method wasn't the best, but it was pretty good.

dlw: I wonder if there were randomization over 1. Strategy or 2. the transformed utiles how it would impact the results? In both cases, it seems the number of candidates approved by voters would be somewhat indeterminate in ways that wouldn't matter for purely ranked-based voting or matter much for IRV3 that treats the ranked votes as approval votes in the first stage.